Package glassomix. May 30, 2013

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1 Package glassomix May 30, 2013 Type Package Title High dimensional Mixture Graph Models selection Version 1.1 Date Author Anani Lotsi and Ernst Wit Maintainer Anani Lotsi Depends R (>= 3.0.0), mvtnorm, glasso, huge Description The package glassomix provides a general framework for network recovering through a model-based soft clustering. It provides functions for parameter estimation via the EM algorithm for Gaussian graphical mixture models in high dimensional setting. The main function is glasso.mix upon which a model selection is performed. The package estimates the optimum number of mixture components (K) and the tuning parameter (lambda) based on the Extended Bayesian Information Criteria (EBIC) via select.gm function. The graphical structural of the K-networks are also plotted through the function gm.plot License GPL (>= 3) URL NeedsCompilation no Repository CRAN Date/Publication :56:04 1

2 2 glassomix-package R topics documented: glassomix-package glasso.mix gm.plot select.gm summary.glasso.mix summary.select.gm Index 13 glassomix-package High dimensional undirected Graphical Mixture Models selection Description A package for high dimensional Undirected Graphical Mixture Models selection. Details This package provides an implementation of the procedures described in Lotsi and Wit (2013). The main function is glasso.mix. This function performs the graph estimation using glasso and a model selction is performed based on Extended Bayesian Information Criterion through the function select.gm. The graphical structural of the K-subgroup of population of individuals is estimated and plotted via the function gm.plot. Functions: glasso.mix Inference via EM algorithm based on glasso. summary.glasso.mix Summary of the result according to fuction glasso.mix select.gm This function performs the model selection and selects the best graphical models based on EBIC. summary.select.gm Summary of the result according to select.gm gm.plot Plot the resultant graphical structural of the K- subgroup of population of individuals. Author(s) Anani Lotsi <a.lotsi@rug.nl> and Ernst Wit

3 glasso.mix 3 References 1. Anani, Lotsi. and E. Wit (2012). High dimensional sparse Gaussian graphical model. (to be published) 2. Witten, Daniela M. and Friedman, Jerome H. and Simon, Noah (2011). New Insights and Faster Computations for the Graphical Lasso. Journal of Computational and Graphical Statistics. 20(4), Pan, Wei and Shen, Xiaotong (2007). Penalized Model-Based Clustering with Application to Variable Selection J. Mach. Learn. Res. 8( ). glasso.mix sparse Gaussian undirected graphical mixture model estimation. Description The main function perfoming the inference via EM algorithm. This function for each value of K, estimate the responsibility matrices ( n K) at the E-step and then given these probabilities, estimate the precision matrices at the M-step via glasso. Usage glasso.mix(data,k=null,lambda=null,em.iter,n.lambda, penalize.diagonal=true,ebic.gamma=0.5,kmax) Arguments data K lambda em.iter ( n p), rows = n, number of observation, columns = p, number of graph nodes/variables). A sequence of integers denoting the numbers of mixture components (clusters) for which EBIC is to be calculated. (Non-negative) regularization parameter for glasso. lambda=0 means no regularization. It could be a scalar or a vector. The maximun number of EM iteration. n.lambda The length of the tuning parameter lambda. penalize.diagonal Should diagonal of precision matrice be penalized? Dafault is FALSE. ebic.gamma The Extended Bayesian Information Criteria paremeter, usually ebic.gamma is between 0 and 1. Kmax The maximum number of K. Details Implements the EM algorithm for a parameterized Gaussian graphical mixture models accross K for each of the regularization parameters.

4 4 glasso.mix Value The details of the output components are as follows: res lambda Kmax n.lambda data A list with the following components: loglik A vector value of un-penalized log-likelihood for each value of K. naiveloglik A vector value of naive log-likelihood extracted from glasso for each value of K. n.par Total number of estimated parameters in each the precision matrices corresponding to each value of K at the various regularization parameters. bestlambda.ebic Optimal tuning parameter corresponding to each value of K. besttheta.ebic The penalized precision matrix corresponding to the optimal EBIC for each value of K. bestpi.ebic The mixture proportion corresponding to the optimal EBIC for each value of K. Theta_Pen All the individual penalized precision matrices corresponding to each value of K at the various regularization parameters. Theta_NonPen All the individual non-penalized precision matrices corresponding to each value of K at the various regularization parameters. pi.ind Responsibility matrices ( n K) corresponding to each value of K for the various regularization parameters. It can also be seen as vector of probabilities (w (k) i1,..., w(k) ik ) of individual i belonging to the k classes at penalty λ. pi K- Mixing coefficients for the various regularization parameters. EBIC All EBIC values for each value of K. The sequence of regularization parameters used. The maximun number of mixture components. The length of the tuning parameter lambda. The data matrix. Author(s) Anani Lotsi and Ernst Wit References 1. Anani, Lotsi. and E. Wit (2013). High dimensional sparse Gaussian graphical model. (to be published) 2. Witten, Daniela M. and Friedman, Jerome H. and Simon, Noah (2011). New Insights and Faster Computations for the Graphical Lasso. Journal of Computational and Graphical Statistics. 20(4), Pan, Wei and Shen, Xiaotong (2007). Penalized Model-Based Clustering with Application to Variable Selection J. Mach. Learn. Res. 8( ). 4. C. Fraley, A. E. Raftery, T. B. Murphy and L. Scrucca (2012). mclust Version 4 for R: Normal Mixture Modeling for Model-Based Clustering, Classification, and Density Estimation. Technical Report No. 597, Department of Statistics, University of Washington.

5 gm.plot 5 See Also summary.glasso.mix, select.gm, summary.select.gm, gm.plot Examples #generate data p<-10 n<-100 ## Number of observations ## Number of nodes L1 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") L2 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") Th1<-L1$sigma ## Precision matrix from graph1 Th2<-L2$sigma ## Precision matrix from graph2 pi1 <- 0.5 z<-rbinom(n,1,pi1) x<-null for (i in 1:n){ if (z[i]==1){ x<-rbind(x,rmvnorm(1,rep(0,p),solve(th1))) else { x<-rbind(x,rmvnorm(1,rep(0,p),solve(th2))) ## Output from the main function ret=glasso.mix(x,k=null,lambda=null,em.iter=5,n.lambda=2, penalize.diagonal=true,ebic.gamma=0.5,kmax=3) gm.plot Graphical plot of the K- Networks Description Plots the K- graphical precision matrices corresponding to the optimum EBIC. Usage gm.plot(output) Arguments output It is a list which is the result of select.gm function. Details It shows the graphical representation (dependencies ) of the p-variables in each cluster. Author(s) Anani Lotsi and Ernst Wit

6 6 select.gm References 1. Anani, Lotsi. and E. Wit (2012). High dimensional sparse Gaussian graphical model. (to be published) 2. Witten, Daniela M. and Friedman, Jerome H. and Simon, Noah (2011). New Insights and Faster Computations for the Graphical Lasso. Journal of Computational and Graphical Statistics. 20(4), Pan, Wei and Shen, Xiaotong (2007). Penalized Model-Based Clustering with Application to Variable Selection J. Mach. Learn. Res. 8( ). 4. C. Fraley, A. E. Raftery, T. B. Murphy and L. Scrucca (2012). mclust Version 4 for R: Normal Mixture Modeling for Model-Based Clustering, Classification, and Density Estimation. Technical Report No. 597, Department of Statistics, University of Washington. See Also glasso.mix, summary.glasso.mix, select.gm, summary.select.gm Examples #generate data p<-10 n<-100 ## Number of observations ## Number of nodes L1 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") L2 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") Th1<-L1$sigma ## Precision matrix from graph1 Th2<-L2$sigma ## Precision matrix from graph2 pi1 <- 0.5 z<-rbinom(n,1,pi1) x<-null for (i in 1:n){ if (z[i]==1){ x<-rbind(x,rmvnorm(1,rep(0,p),solve(th1))) else { x<-rbind(x,rmvnorm(1,rep(0,p),solve(th2))) ret=glasso.mix(x,k=null,lambda=null,em.iter=5,n.lambda=2, penalize.diagonal=true,ebic.gamma=0.5,kmax=3) output=select.gm(ret) graph=gm.plot(output) select.gm High dimensional sparse Gaussian graphical mixture model selection

7 select.gm 7 Description This function selects the optimal model according to Extended Bayesian Information Criterin (EBIC) for EM- algorithm for parameterized High dimensional sparse Gaussian graphical mixture models. The function etimates the optimun number of mixture components and the regularization parameter lambda. Usage select.gm(ret) Arguments ret It is a list which is the result of glasso.mix algorithm from glasso.mix function. Details Implements the model selection clustering through a model selection based on the EBIC for a parameterized Gaussian graphical mixture model accross K for each of the regularization parameters. Value The details of the output components are as follows: n.cluster Optimal number of clusters or mixture components K. ebic lambda_ebic Th.Pen Th.NPen Pi_ind Pi clusters Pen_LogLik NPen_LogLik lambda All EBIC values. This is a matrix with row equals maximun number of mixtures minus one; (Kmax-1) and column equals to n.lambda. Optimum lambda value based on EBIC. Optimum penalized K precision matrices. Optimum non-penalized K precision matrices. Optimum responsibility matrices ( n K) corresponding to the soft-k-means clustering. Optimum (K) mixture proportions based on EBIC criterion. ( n 1) vector containing the indices of the clusters where the data points are assigned to. The un-penalized loglikelihood corresponding to the optimal EBIC. The naive un-penalized loglikelihood corresponding to the optimal EBIC. The sequence of regularization parameters used. Author(s) Anani Lotsi and Ernst Wit

8 8 summary.glasso.mix References 1. Anani, Lotsi. and E. Wit (2012). High dimensional sparse Gaussian graphical model. (to be published) 2. Witten, Daniela M. and Friedman, Jerome H. and Simon, Noah (2011). New Insights and Faster Computations for the Graphical Lasso. Journal of Computational and Graphical Statistics. 20(4), Pan, Wei and Shen, Xiaotong (2007). Penalized Model-Based Clustering with Application to Variable Selection J. Mach. Learn. Res. 8( ). 4. C. Fraley, A. E. Raftery, T. B. Murphy and L. Scrucca (2012). mclust Version 4 for R: Normal Mixture Modeling for Model-Based Clustering, Classification, and Density Estimation. Technical Report No. 597, Department of Statistics, University of Washington. See Also glasso.mix, summary.glasso.mix, summary.select.gm, gm.plot Examples #generate data p<-10 n<-100 ## Number of observations ## Number of nodes L1 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") L2 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") Th1<-L1$sigma ## Precision matrix from graph1 Th2<-L2$sigma ## Precision matrix from graph2 pi1 <- 0.5 z<-rbinom(n,1,pi1) x<-null for (i in 1:n){ if (z[i]==1){ x<-rbind(x,rmvnorm(1,rep(0,p),solve(th1))) else { x<-rbind(x,rmvnorm(1,rep(0,p),solve(th2))) ret=glasso.mix(x,k=null,lambda=null,em.iter=5,n.lambda=2, penalize.diagonal=true,ebic.gamma=0.5,kmax=3) output=select.gm(ret) summary.glasso.mix Summary according to function glasso.mix Description Reduced summary of the result according to glasso.mix

9 summary.glasso.mix 9 Usage ## S3 method for class glasso.mix summary(object,...) Arguments Details Value object an object with S3 class glasso.mix. A list of the result from the function glasso.mix function.... system reserved (no specific usage). It gives a reduced summary of output from glasso.mix. The details of the output components are as follows: lambda The sequence of regularization parameters. Pi Mixture proportions for each K across lambda. bestlambda.ebic Optimum lambda value based on EBIC for each K. besttheta.ebic The penalized precision matrix corresponding to the optimal EBIC for each value of K. n.par Author(s) Anani Lotsi and Ernst Wit References Total number of estimated parameters in the precision matrices corresponding to each value of K at the various regularization parameters. 1. Anani, Lotsi. and E. Wit (2012). High dimensional sparse Gaussian graphical model. (to be published) 2. Witten, Daniela M. and Friedman, Jerome H. and Simon, Noah (2011). New Insights and Faster Computations for the Graphical Lasso. Journal of Computational and Graphical Statistics. 20(4), Pan, Wei and Shen, Xiaotong (2007). Penalized Model-Based Clustering with Application to Variable Selection J. Mach. Learn. Res. 8( ), ( ) 4. C. Fraley, A. E. Raftery, T. B. Murphy and L. Scrucca (2012). mclust Version 4 for R: Normal Mixture Modeling for Model-Based Clustering, Classification, and Density Estimation. Technical Report No. 597, Department of Statistics, University of Washington. See Also glasso.mix, select.gm, summary.select.gm, gm.plot

10 10 summary.select.gm Examples #generate data p<-10 n<-100 ## Number of observations ## Number of nodes L1 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") L2 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") Th1<-L1$sigma ## Precision matrix from graph1 Th2<-L2$sigma ## Precision matrix from graph2 pi1 <- 0.5 z<-rbinom(n,1,pi1) x<-null for (i in 1:n){ if (z[i]==1){ x<-rbind(x,rmvnorm(1,rep(0,p),solve(th1))) else { x<-rbind(x,rmvnorm(1,rep(0,p),solve(th2))) ret=glasso.mix(x,k=null,lambda=null,em.iter=5,n.lambda=2, penalize.diagonal=true,ebic.gamma=0.5,kmax=3) summary.glasso.mix(ret) summary.select.gm Summary according to the model selection function select.gm Description Summary of the result according to select.gm Usage ## S3 method for class select.gm summary(object,...) Arguments object an object with S3 class select.gm. A list of the result from the select.gm function.... system reserved (no specific usage). Details It gives summary of output from select.gm.

11 summary.select.gm 11 Value The details of the output components are as follows: mix_comp Optimal number of clusters mixture components K. lambda_ebic clustering mix_prop Author(s) Anani Lotsi and Ernst Wit References Optimum lambda value based on EBIC. ( n 1) vector containing the indices of the clusters where the data points are assigned to. Optimum (K) mixture proportions. 1. Anani, Lotsi. and E. Wit (2012). High dimensional sparse Gaussian graphical model. (to be published) 2. Witten, Daniela M. and Friedman, Jerome H. and Simon, Noah (2011). New Insights and Faster Computations for the Graphical Lasso. Journal of Computational and Graphical Statistics. 20(4), Pan, Wei and Shen, Xiaotong (2007). Penalized Model-Based Clustering with Application to Variable Selection J. Mach. Learn. Res. 8( ), ( ) 4. C. Fraley, A. E. Raftery, T. B. Murphy and L. Scrucca (2012). mclust Version 4 for R: Normal Mixture Modeling for Model-Based Clustering, Classification, and Density Estimation. Technical Report No. 597, Department of Statistics, University of Washington. See Also glasso.mix, summary.glasso.mix, select.gm, gm.plot Examples p<-10 n<-100 ## Number of observations ## Number of nodes L1 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") L2 = huge.generator(n = n, d = p, vis = FALSE, graph = "random") Th1<-L1$sigma ## Precision matrix from graph1 Th2<-L2$sigma ## Precision matrix from graph2 pi1 <- 0.5 z<-rbinom(n,1,pi1) x<-null for (i in 1:n){ if (z[i]==1){ x<-rbind(x,rmvnorm(1,rep(0,p),solve(th1))) else { x<-rbind(x,rmvnorm(1,rep(0,p),solve(th2)))

12 12 summary.select.gm ret=glasso.mix(x,k=null,lambda=null,em.iter=5,n.lambda=2, penalize.diagonal=true,ebic.gamma=0.5,kmax=3) output=select.gm(ret) summary.select.gm(output)

13 Index Topic package glassomix-package, 2 glasso.mix, 3, 6, 8, 9, 11 glassomix (glassomix-package), 2 glassomix-package, 2 gm.plot, 5, 5, 8, 9, 11 select.gm, 5, 6, 6, 9, 11 summary.glasso.mix, 5, 6, 8, 8, 11 summary.select.gm, 5, 6, 8, 9, 10 13